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It is known that Horndeski theories can be transformed to a sub-class ofGleyzes-Langlois-Piazza-Vernizzi (GLPV) theories under the disformaltransformation of the metric $g_{\mu \nu} \to \Omega^2(\phi)g_{\mu \nu}+\Gamma(\phi,X) \nabla_{\mu} \phi \nabla_{\nu} \phi$, where $\Omega$ is a function ofa scalar field $\phi$ and $\Gamma$ is another function depending on both $\phi$and $X=g^{\mu \nu}\nabla_{\mu} \phi \nabla_{\nu} \phi$. We show that, with thechoice of unitary gauge, both curvature and tensor perturbations on the flatisotropic cosmological background are generally invariant under the disformaltransformation. By means of the effective field theories encompassing Horndeskiand GLPV theories, we obtain the second-order actions of scalar/tensorperturbations and present the relations for physical quantities between the twoframes. The invariance of the inflationary power spectra under the disformaltransformation is explicitly proved up to next-to-leading order in slow-roll.In particular, we identify the existence of the Einstein frame in which thetensor power spectrum is of the same form as that in General Relativity andderive the condition under which the spectrum of gravitational waves in GLPVtheories is red-tilted.

Disformal invariance of cosmological perturbations in a generalized class of Horndeski theories